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A159599 E.g.f.: A(x) = exp( Sum_{n>=1} [ D^n exp(x) ]^n/n ), where differential operator D = x*d/dx.

%I A159599 #4 Sep 14 2013 14:08:42
%S A159599 1,1,4,27,304,5685,177486,9305821,807656872,113141689065,
%T A159599 25091265489130,8644033129800321,4584172093683770820,
%U A159599 3704744323753306881229,4538175408875808587259022,8381136688938251234193247485
%N A159599 E.g.f.: A(x) = exp( Sum_{n>=1} [ D^n exp(x) ]^n/n ), where differential operator D = x*d/dx.
%F A159599 E.g.f.: A(x) = exp( Sum_{n>=1} [ Sum_{k>=1} k^n*x^k/k! ]^n/n ).
%e A159599 E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 304*x^4/4! +...
%e A159599 log(A(x)) = x + 3*x^2/2! + 17*x^3/3! + 190*x^4/4! + 3889*x^5/5! +...
%e A159599 log(A(x)) = (D^1 e^x) + (D^2 e^x)^2/2 + (D^3 e^x)^3/3 +...
%e A159599 D^1 exp(x) = (1)*x*exp(x);
%e A159599 D^2 exp(x) = (1 + x)*x*exp(x);
%e A159599 D^3 exp(x) = (1 + 3*x + x^2)*x*exp(x);
%e A159599 D^4 exp(x) = (1 + 7*x + 6*x^2 + x^3)*x*exp(x);
%e A159599 D^5 exp(x) = (1 + 15*x + 25*x^2 + 10*x^3 + x^4)*x*exp(x); ...
%e A159599 D^n exp(x) = n-th iteration of operator D = x*d/dx on exp(x) equals the g.f. of the n-th row of triangle A008277 (S2(n,k)) times x*exp(x), and so is related to the n-th Bell number.
%o A159599 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n, sum(k=1,n,k^m*x^k/k!+x*O(x^n))^m/m))); n!*polcoeff(A,n)}
%Y A159599 Cf. A159596, A008277 (S2(n, k)), A000110 (Bell).
%K A159599 nonn
%O A159599 0,3
%A A159599 _Paul D. Hanna_, May 05 2009, May 22 2009